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Photon wave function, electric and magnetic fields. Relationship to the electromagnetic vacuum.

Constantin Meis

Published May 22, 2024• 0 citations
Publisher

Abstract

A photon quantization volume is defined, an intrinsic physical geometrical property of a single photon state, showing that photons are not point particles and constitute a particular case in the standard particle model. Based on the quantized vector potential amplitude a photon wave function is established satisfying Schrödinger’s equation and normalized with respect to the quantization volume. The single photon electric and magnetic fields in vacuum are calculated independently on any external cavity parameter and they are proportional to the square of the frequency. The electromagnetic vacuum derives straightforward from the photon vector potential function and has both classical and quantum representations. The photon wave function is directly related to the electromagnetic vacuum states setting the fundamentals for understanding entanglement.

DOI
10.1364/opticaopen.25868590.v1

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A photon quantization volume is defined, an intrinsic physical geometrical property of a single photon state, showing that photons are not point particles and constitute a particular case in the standard particle model. Based on the quantized vector potential amplitude a photon wave function is established satisfying Schrödinger’s equation and normalized with respect to the quantization volume. The single photon electric and magnetic fields in vacuum are calculated independently on any external cavity parameter and they are proportional to the square of the frequency. The electromagnetic vacuum derives straightforward from the photon vector potential function and has both classical and quantum representations. The photon wave function is directly related to the electromagnetic vacuum states setting the fundamentals for understanding entanglement.
A photon quantization volume is defined, an intrinsic physical geometrical property of a single photon state, showing that photons are not point particles and constitute a particular case in the standard particle model. Based on the quantized vector potential amplitude a photon wave function is established satisfying Schrödinger’s equation and normalized with respect to the quantization volume. The single photon electric and magnetic fields in vacuum are calculated independently on any external cavity parameter and they are proportional to the square of the frequency. The electromagnetic vacuum derives straightforward from the photon vector potential function and has both classical and quantum representations. The photon wave function is directly related to the electromagnetic vacuum states setting the fundamentals for understanding entanglement.
We employ here the enhancement of the vector potential amplitude quantization at a single photon state. The analysis of the general solution of the vector potential, obtained by resolving Maxwell’s equations, implies that the amplitude is proportional to the angular frequency. The photon vector potential function αkλ(r,t) can be written in the plane wave representation satisfying the classical wave propagation equation, Schrödinger’s equation for the energy with the relativistic massless field Hamiltonian and a linear time-dependent equation for the vector potential amplitude operator. Thus, the vector potential αkλ(r,t) with the quantized amplitude may play the role of a real wave function for the photon in a nonlocal representation that can be suitably normalized. We then deduce that the amplitudes of the electric and magnetic fields, respectively, of a single free photon are proportional to the square of the angular frequency. This might open perspectives for the development of nondestructive photon detection methods based on the influence of the electric and/or magnetic fields of photons on the energy levels of atoms and molecules.
The quantization procedure of the vector potential is enhanced at a single photon state revealing the possibility for a simultaneous representation of the wave-particle nature of the photon. Its relationship to the quantum vacuum results naturally. A vector potential amplitude operator is defined showing the parallelism with the Hamiltonian of a massless particle. It is further shown that the quantized vector potential satisfies both the wave propagation equation and a linear time-dependent Schrödinger-like equation.
The vector potential function ) , ( t r k & & O 0 2H Z k we define a six components function as a general function for a k-mode photon.The square of the modulus of the defined general function gives the energy density at a given coordinate which depends on the fourth power of the angular frequency.The amplitudes of the electric and magnetic fields of a single k-mode photon free of cavity are also calculated and it is shown that they are proportional to the square of the angular frequency.In this way, the influence of the photon electric and/or magnetic fields on the energy levels of atoms and molecules might be used for a non-destructive photon detection.
A photon structure is advanced based on the experimental evidence and the vector potential quantization at a single photon level. It is shown that the photon is neither a point particle nor an infinite wave but behaves rather like a local “wave-corpuscle” extended over a wavelength, occupying a minimum quantization volume and guided by a non-local vector potential real wave function. The quantized vector potential oscillates over a wavelength with circular left or right polarization giving birth to orthogonal magnetic and electric fields whose amplitudes are proportional to the square of the frequency. The energy and momentum are carried by the local wave-corpuscle guided by the non-local vector potential wave function suitably normalized.
Quantum theory of photons based on the first quantization technique, similar to that used by Schrödinger in the formulation of quantum mechanics, is considered. First, scalar quantum mechanics of photons operating with the photon wave functions is discussed. Using the first quantization, the wave equation, the Schrödinger-like equations, and the Dirac equation for photons are derived. Then, vector quantum mechanics of photons is introduced, which defines the electromagnetic vector fields. Using the first quantization, the Maxwell equations for photons in a magneto-dielectric medium are obtained. Because the photon’s electric and magnetic fields satisfy the Maxwell equations, all that is known about the classical optical fields can be directly transferred to photons demonstrating their quantum diversity. Relationships between the scalar and vector quantum mechanics of photons and between the Dirac and Maxwell equations are analyzed. To describe the propagation of photons in dispersive media, modified Maxwell equations are introduced.